Evaluation of Ranks of Real Space and Particle Entanglement Spectra for Large Systems

We devise a way to calculate the dimensions of symmetry sectors appearing in the particle entanglement spectrum (PES) and real space entanglement spectrum (RSES) of multiparticle systems from their real space wave functions. We first note that these ranks in the entanglement spectra equal the dimensions of spaces of wave functions with a number of particles fixed. This also yields equality of the multiplicities in the PES and the RSES. Our technique allows numerical calculations for much larger systems than were previously feasible. For somewhat smaller systems, we can find approximate entanglement energies as well as multiplicities. We illustrate the method with results on the RSES and PES multiplicities for integer quantum Hall states, Laughlin and Jain composite fermion states, and for the Moore-Read state at filling $\nu =\frac{5}{2}$ for system sizes up to 70 particles.

Figure 1

Plot of the spectra of the matrices $M^{†}M$ used in the calculation of the RSES ranks for a $\nu =\frac{1}{2}$ Laughlin state at $N=70$ and $N_A=35$ on a sphere. The negative logarithms of the eigenvalues for each $L_z^A$ block of ${\rho }_A$ are plotted (the largest eigenvalue is normalized to 1 at each $L_z^A$). A clear gap is visible for each value of $L_z^A$ and the counting of large eigenvalues (at low $ϵ$) matches the partition numbers.

Figure 2

Plot of the spectra of the matrices $M^{†}M$ used in the calculation of the RSES ranks for a $\nu =\frac{2}{5}$ Jain state on a sphere at $N=20$ and $N_A=10$ (left) and at $N=22$ and $N_A=11$ (right). Negative values of $ϵ$ occur because we used an unnormalized Jain state (normalization shifts the vertical axis by a constant). The counting $1,2,5,10,\dots$ of the branches is consistent with a pair of noninteracting Luttinger liquids. Differences in total counting for even vs odd $N_A$ arise analogously to those for $\nu =2$.